Convicted killer Nathan Dunlap lost his last, best chance to avoid execution Tuesday when the United States Supreme Court refused to hear his appeal.

Dunlap was sentenced to die for killing four people in 1993 in an Aurora Chuck E. Cheese restaurant. He is Colorado's longest-serving death-row inmate, and could become the first inmate executed in the state since 1997.

A list of unsigned orders issued Tuesday morning by the nation's highest court shows that Dunlap's appeal to the court challenging his death sentence -- known as a petition for certiorari -- was denied. That ends the last appeal Dunlap is guaranteed under the law and clears the way for an execution date to be set.

Dunlap, 38, may file further appeals, but they are not certain to delay execution.

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In a statement to The Denver Post, one of Dunlap's attorneys expressed disappointment in the development.

"Mr. Dunlap should spend the rest of his life in prison without the possibility of parole," attorney Phil Cherner said. "What happened is tragic. But taking his life isn't going to change that. We wish it were different, but it isn't. However, given what we know about the unfair and disproportionate use of capital punishment in Colorado, it would be unconscionable for the state to carry out this death sentence."

The next step in the process is for the 18th Judicial District Attorney's office, which prosecuted the case, to make a motion to an Arapahoe County District Court judge to issue a death warrant. According to state law, the death warrant will specify a week during which Dunlap should be executed. It is up to the head of the state Department of Corrections to pick the exact day of execution.

According to a Corrections Department regulation, no one at the department will be allowed to say publicly when the execution date is.

Executions must be carried out by lethal injection under Colorado law.

Dunlap was 19 when he killed four employees of a Chuck E. Cheese restaurant where he used to work. Three of the victims were teenagers: 19-year-old Sylvia Crowell and 17-year-olds Ben Grant and Colleen O'Connor. Another, 50-year-old Margaret Kohlberg, was the restaurant's manager. Dunlap also gravely wounded another employee, who was able to escape and find help.

Police said Dunlap -- who has now spent half his life behind bars for the killings -- was upset over being fired from the restaurant.

Dunlap was convicted and sentenced to death in 1996, and his sentence subsequently withstood a series of appeals at the same time several other Colorado death-row inmates saw their sentences overturned.

Last year, Dunlap's attorneys argued before the 10th Circuit Court of Appeals that Dunlap's death sentence should be thrown out because of alleged incompetence by his trial lawyers. Those lawyers, Dunlap's appellate attorneys argued, failed to introduce evidence at his sentencing hearing that Dunlap is mentally ill. The appellate attorneys said that, had jurors heard such evidence, at least one might have refused a death sentence.

A panel of 10th Circuit judges, though, rejected the argument, saying that Dunlap's trial lawyers did the best job they could with a difficult case.

Dunlap's ongoing appeals have been difficult for family members of the victims. Before last year's 10th Circuit hearing, Sylvia Crowell's mother, Marj, told The Denver Post that she expected the hearing to be "like salt in the wound."

"That hurt is still going on," Bob Crowell, Sylvia's father, said then. "And we are somewhat anxious that somebody is going to throw a monkey wrench in there and he is not going to be executed."

The last inmate executed in Colorado -- murderer Gary Lee Davis -- was killed five months after the U.S. Supreme Court declined to hear his last legally guaranteed appeal. Davis decided not to file additional appeals.

Dunlap's attorneys have not said whether they plan further appeals. If they do, one possibility is challenging the constitutionality of Colorado's death-penalty laws.

Attorneys for two-time killer Edward Montour have tried a similar approach in seeking to keep Montour from a looming death-penalty sentencing hearing. A study by two University of Denver law professors questions whether Colorado's law is unconstitutionally arbitrary because prosecutors pursue capital punishment at trial in only about 1 percent of death-eligible cases.

Aside from another appeal, Dunlap's attorneys could also petition Gov. John Hickenlooper for clemency.